Broadband Müller-MLFMA for Electromagnetic Scattering by Dielectric Objects

The multilevel fast multipole algorithm (MLFMA) is very efficient for solving large-scale electromagnetic scattering problems. However, at low frequencies, or when the discretization is small compared with the wavelength, both the MLFMA and the underlying integral equation formulation typically suffer from a subwavelength breakdown. For the electromagnetic scattering from a homogeneous dielectric object, we obtain a stable and well-conditioned surface integral formulation using a variant of the classical Muumlller formulation and linear basis functions. To overcome the subwavelength breakdown of the MLFMA, we use both propagating and evanescent plane waves to represent the fields. The implementation is based on a combination of the spectral representation of the Green's function and Rokhlin's translation formula. We also present a new interpolation scheme for the evanescent part, which significantly improves the error-controllability of the MLFMA-implementation. Several numerical results verify both the error-controllability and scalability of the proposed algorithm     

Improving Conditioning of Electromagnetic Surface Integral Equations Using Normalized Field Quantities

When the surface integral equation method is applied to study electromagnetic scattering by dielectric or composite metallic and dielectric objects, the unknowns, i.e., the electric and magnetic surface current densities, and the elements of the system matrix, are often of the very different scales. As a consequence, the system matrix may have a high (singular value) condition number. An efficient method is presented to balance the unknowns and the integral equations, and the elements of the system matrix, too. The method is based on the use of normalized field quantities and unknowns, and carefully chosen scaling factors. In the case of dielectric and composite objects the condition numbers of the SIE matrices can be reduced with several orders of magnitudes by the developed method. In the case of high contrast objects, or if the frequency is very low, the developed method leads also to a clear improvement on the convergence of iterative solutions     

Polarizabilities of platonic solids

This article presents results of a numerical effort to determine the dielectric polarizabilities of the five regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The polarizability is calculated by solving a surface integral equation, in which the unknown potential is expanded using third-order basis functions. The resulting polarizabilities are accurate to the order of 10/sup -4/. Approximation formulas are given for the polarizabilities as functions of permittivity. Among other results, it is found that the polarizability of a regular polyhedron correlates more strongly with the number of edges than with the number of faces, vertices, or the solid angle seen from a vertex.     

Current and charge Integral equation formulation

A new stable frequency domain surface integral equation formulation is proposed for the three dimensional electromagnetic scattering of composite metallic and dielectric objects. The developed formulation does not suffer from the low frequency breakdown and leads to a well balanced and stable system on a wide frequency band. Surface charge densities are used as unknowns in addition to the traditional surface current densities. The balance of the system is achieved by using normalized field quantities and by enforcing the continuity of the fields across the boundaries with carefully chosen scaling factors. The linear dependence between the currents and charges is taken into account with an integral operator, and the linear dependence in charges is removed with the deflation method. A combined field integral equation form of the formulation is proposed to remove the internal resonance problem associated to the closed metallic objects. Due to the good balance in the new formulation, fast converging iterative solutions on a very wide frequency band can be obtained. The new formulation and its convergence is verified with numerical examples.     

Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles

In this paper a singularity subtraction technique is developed for computing the impedance matrix elements of various electromagnetic surface integral equation formulations with the Galerkin method and high-order basis functions. Analytical closed form formulas for computing surface integrals with |r - r'|/sup n/, n/spl ges/-3, singularities times polynomial nodal shape functions of arbitrary order on a planar triangle are presented.     

Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions

The method of moments (MoM) solution of combined field integral equation (CFIE) for electromagnetic scattering problems requires calculation of singular double surface integrals. When Galerkin's method with triangular vector basis functions, Rao-Wilton-Glisson functions, and the CFIE are applied to solve electromagnetic scattering by a dielectric object, both RWG and n/spl times/RWG functions (n is normal unit vector) should be considered as testing functions. Robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the CFIE with these basis and test functions. In computing the impedance matrix elements, including the gradient of the Green's function, we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand. In the developed method, all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions. In addition, we present compact iterative formulas for computing the extracted terms in closed form. By these formulas, we can extract any number of terms from the singular kernels of CFIE formulations with RWG and n/spl times/RWG functions.     

Zero Backscattering From Self-Dual Objects of Finite Size

Duality transformation is applied to the theory of zero backscattering from finite objects. It is shown that if the object, defined by medium and/or boundary condition, is self dual, i.e., invariant in the duality transformation, it is invisible to radar if a certain condition for the polarizability dyadic is valid. This is a general statement and includes previous theorems as special cases. As novel self-dual objects those with boundary conditions requiring vanishing of the normal components of the D and B vectors ("DB boundary") or their normal derivatives ("D^'B^' boundary") are introduced. As examples of zero-backscattering objects, scattering properties of a sphere and a cube with DB or D^'B^' boundary, as well as an anisotropic asymmetric spheroid, are discussed.